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Theorem sqabssub 10768
Description: Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.)
Assertion
Ref Expression
sqabssub  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( ( ( abs `  A
) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( A  x.  ( * `  B
) ) ) ) ) )

Proof of Theorem sqabssub
StepHypRef Expression
1 cjsub 10604 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  -  B )
)  =  ( ( * `  A )  -  ( * `  B ) ) )
21oveq2d 5756 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  x.  (
* `  ( A  -  B ) ) )  =  ( ( A  -  B )  x.  ( ( * `  A )  -  (
* `  B )
) ) )
3 cjcl 10560 . . . . 5  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
4 cjcl 10560 . . . . 5  |-  ( B  e.  CC  ->  (
* `  B )  e.  CC )
53, 4anim12i 334 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  e.  CC  /\  ( * `  B
)  e.  CC ) )
6 mulsub 8127 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( * `
 A )  e.  CC  /\  ( * `
 B )  e.  CC ) )  -> 
( ( A  -  B )  x.  (
( * `  A
)  -  ( * `
 B ) ) )  =  ( ( ( A  x.  (
* `  A )
)  +  ( ( * `  B )  x.  B ) )  -  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) ) )
75, 6mpdan 415 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  x.  (
( * `  A
)  -  ( * `
 B ) ) )  =  ( ( ( A  x.  (
* `  A )
)  +  ( ( * `  B )  x.  B ) )  -  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) ) )
82, 7eqtrd 2148 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  -  B )  x.  (
* `  ( A  -  B ) ) )  =  ( ( ( A  x.  ( * `
 A ) )  +  ( ( * `
 B )  x.  B ) )  -  ( ( A  x.  ( * `  B
) )  +  ( ( * `  A
)  x.  B ) ) ) )
9 subcl 7925 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  B
)  e.  CC )
10 absvalsq 10765 . . 3  |-  ( ( A  -  B )  e.  CC  ->  (
( abs `  ( A  -  B )
) ^ 2 )  =  ( ( A  -  B )  x.  ( * `  ( A  -  B )
) ) )
119, 10syl 14 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( A  -  B )  x.  ( * `  ( A  -  B )
) ) )
12 absvalsq 10765 . . . 4  |-  ( A  e.  CC  ->  (
( abs `  A
) ^ 2 )  =  ( A  x.  ( * `  A
) ) )
13 absvalsq 10765 . . . . 5  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( B  x.  ( * `  B
) ) )
14 mulcom 7713 . . . . . 6  |-  ( ( B  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( B  x.  ( * `  B
) )  =  ( ( * `  B
)  x.  B ) )
154, 14mpdan 415 . . . . 5  |-  ( B  e.  CC  ->  ( B  x.  ( * `  B ) )  =  ( ( * `  B )  x.  B
) )
1613, 15eqtrd 2148 . . . 4  |-  ( B  e.  CC  ->  (
( abs `  B
) ^ 2 )  =  ( ( * `
 B )  x.  B ) )
1712, 16oveqan12d 5759 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  =  ( ( A  x.  ( * `  A ) )  +  ( ( * `  B )  x.  B
) ) )
18 mulcl 7711 . . . . . 6  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( A  x.  ( * `  B
) )  e.  CC )
194, 18sylan2 282 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
* `  B )
)  e.  CC )
2019addcjd 10669 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )
21 cjmul 10597 . . . . . . 7  |-  ( ( A  e.  CC  /\  ( * `  B
)  e.  CC )  ->  ( * `  ( A  x.  (
* `  B )
) )  =  ( ( * `  A
)  x.  ( * `
 ( * `  B ) ) ) )
224, 21sylan2 282 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  ( * `  (
* `  B )
) ) )
23 cjcj 10595 . . . . . . . 8  |-  ( B  e.  CC  ->  (
* `  ( * `  B ) )  =  B )
2423adantl 273 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  (
* `  B )
)  =  B )
2524oveq2d 5756 . . . . . 6  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( * `  A )  x.  (
* `  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2622, 25eqtrd 2148 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( * `  ( A  x.  ( * `  B ) ) )  =  ( ( * `
 A )  x.  B ) )
2726oveq2d 5756 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( * `  B
) )  +  ( * `  ( A  x.  ( * `  B ) ) ) )  =  ( ( A  x.  ( * `
 B ) )  +  ( ( * `
 A )  x.  B ) ) )
2820, 27eqtr3d 2150 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
Re `  ( A  x.  ( * `  B
) ) ) )  =  ( ( A  x.  ( * `  B ) )  +  ( ( * `  A )  x.  B
) ) )
2917, 28oveq12d 5758 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( abs `  A ) ^ 2 )  +  ( ( abs `  B
) ^ 2 ) )  -  ( 2  x.  ( Re `  ( A  x.  (
* `  B )
) ) ) )  =  ( ( ( A  x.  ( * `
 A ) )  +  ( ( * `
 B )  x.  B ) )  -  ( ( A  x.  ( * `  B
) )  +  ( ( * `  A
)  x.  B ) ) ) )
308, 11, 293eqtr4d 2158 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( abs `  ( A  -  B )
) ^ 2 )  =  ( ( ( ( abs `  A
) ^ 2 )  +  ( ( abs `  B ) ^ 2 ) )  -  (
2  x.  ( Re
`  ( A  x.  ( * `  B
) ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   ` cfv 5091  (class class class)co 5740   CCcc 7582    + caddc 7587    x. cmul 7589    - cmin 7897   2c2 8728   ^cexp 10232   *ccj 10551   Recre 10552   abscabs 10709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-mulrcl 7683  ax-addcom 7684  ax-mulcom 7685  ax-addass 7686  ax-mulass 7687  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-1rid 7691  ax-0id 7692  ax-rnegex 7693  ax-precex 7694  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700  ax-pre-mulgt0 7701  ax-pre-mulext 7702  ax-arch 7703  ax-caucvg 7704
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rmo 2399  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-po 4186  df-iso 4187  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-reap 8300  df-ap 8307  df-div 8393  df-inn 8678  df-2 8736  df-3 8737  df-4 8738  df-n0 8929  df-z 9006  df-uz 9276  df-rp 9391  df-seqfrec 10159  df-exp 10233  df-cj 10554  df-re 10555  df-im 10556  df-rsqrt 10710  df-abs 10711
This theorem is referenced by:  sqabssubi  10865
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